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Cubic Twin Prime Polynomials are Counted by a Modular Form

  • Lior Bary-Soroker (a1) and Jakob Stix (a2)

We present the geometry behind counting twin prime polynomials in $\mathbb{F}_{q}[T]$ in general. We compute cohomology and explicitly count points by means of a twisted Lefschetz trace formula applied to these parametrizing varieties for cubic twin prime polynomials. The elliptic curve $X^{3}=Y(Y-1)$ occurs in the geometry, and thus counting cubic twin prime polynomials involves the associated modular form. In theory, this approach can be extended to higher degree twin primes, but the computations become harder.

The formula we get in degree 3 is compatible with the Hardy–Littlewood heuristic on average, agrees with the prediction for $q\equiv 2$ (mod 3), but shows anomalies for $q\equiv 1$ (mod 3).

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The authors acknowledge support provided by DAAD-Programm 57271540 Strategische Partnerschaften (supported by BMBF). The first author was partially supported by a grant from the Israel Science Foundation.

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Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
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